Residence Times

Question 1

Define ‘residence time’.

Answer 1

The actual time the fluid resides within the rector.

Question 2

How do you calculate residence time?

Answer 2

t = V/u(t)
u(t) = volumetric flow rate.

Question 3

For a batch reactor, how can residence be calculated generally?

Answer 3

residence time = reaction time

Question 4

Define ‘space time’.

Answer 4

The residence time when volumetric flow rate is constant throughout.

Question 5

How do you calculate space velocity?

Answer 5

Sv = u(t)/V

-> inverse of space time.

Question 6

How are residence times used to diagnose reactor problems?

Answer 6

An inert tracer is injected into the reactor inlet and the outlet concentration is continuously monitored.

Question 7

What does a short residence time depict about a reactor?

Answer 7

By-passing -> reactants are not seeing the full volume of the reactor eg. short-circuiting & channeling

Question 8

What does a longer residence time depict about a reactor?

Answer 8

Calculation error -> incorrect volume or volumetric flow eg. inlet volume not considered.
Poor mixing -> stagnant regions & short-circuiting.

Question 9

Derive the formula for the time taken for a reaction to occur in a batch reactor.

Answer 9

rA V = - dNA/dt

1 dt = NA0/rA V

Question 10

For a liquid phase reaction, derive the residence time in a batch reactor.
n=reaction order

Answer 10

1 dt = NA0/rA V

rA = kCA = kNA/V = k ((NA(1-x))/V)

t = 1/kNA^(n-1) x V^(n-1) x dxA/(1-x)^n

Question 11

For a liquid-phase reaction with constant volume, derive the residence time in a batch reactor.
n=reaction order

Answer 11

t = 1/kNA^(n-1) x V^(n-1) x dxA/(1-x)^n

t = V^(n-1)/kNA^(n-1) x dxA/(1-x)^n

t = 1/kCA0^(n-1) x dxA/(1-x)^n

Question 12

What is the equation for a first order, constant volume, gas phase reaction in a batch reactor?

Answer 12

t = -1/k ln(1-xA)

Question 13

For a gas-phase reaction with constant pressure, derive the residence time for a batch reactor.

Answer 13

t = 1/kNA^(n-1) x V^(n-1) x dxA/(1-x)^n

NT = NT0 + ((vB+vC-vA)/vA) NA0 xA

NT/NT0 = 1 + eA xA

Ideal gas; V/V0 = NT/NT0 = 1 + eA xA
-> V = V0(1+eA xA)

t = 1/kCA0^(n-1) x (1+eA xA)^(n-1)/(1-xA)^n x dxA

Question 14

Derive the formula for the volume required in a PFR.

Answer 14

rA = -dnA/dV

1 dV = nA0/rA dxA

Question 15

For a liquid-phase reaction, derive the residence volume occupied in a PFR.

Answer 15

1 dV = nA0/rA dxA

rA = kCA = knA/u(t) = k nA0^n (1-xA)^n/u(t)^n

u(t) = u(t0) (1+eAxA)

V = u(t0)/knA0^(n-1) x (1+eAxA)/(1-xA)^n dxA

Question 16

For a constant volumetric flowrate, liquid-phase reaction, what is the residence volume occupied?

Answer 16

V = u(t0)/knA0^(n-1) x 1/(1-xA)^n dxA

Question 17

What is the equation for the residence volume in a first order, liquid phase reaction in a PFR?

Answer 17

V = u(t)/k ln(1/(1-xA))

Question 18

What is the equation for conversion of A in a first order, gas phase reaction in a PFR?

Answer 18

xA = 1 - e^(-kt)

Question 19

Derive the equation for the volume of a CSTR from reaction rate.

Answer 19

rA = (nA0-nA)/V
V = (nA0 xA)/rA

Question 20

Derive the equation for the residence volume in a CSTR.

Answer 20

V = (nA0 xA)/rA

rA = kCA^n = k (nA/u(t))^n

u(t) = u(t0) (1+eAxA)

V = (nA0/k) x (1/CA0^n) x ((1+eAxA)^n/(1-xA)^n) x xA

Question 21

For a first order reaction, define an equation for the mean residence time in a CSTR.

Answer 21

t = V/u(t) = 1/k x xA/(1-xA)

Question 22

For a first order reaction, define an equation for the conversion of A in a CSTR.

Answer 22

xA = kt/(1+kt)

Question 23

Why is the residence time defined as the mean residence time in a CSTR?

Answer 23

Not all molecules in a CSTR have the same residence time but instead are a distribution.

Question 24

What does a shorter mean residence time depict in a CSTR?

Answer 24

At any one time, molecules will have been in the reactor for a range of residence times so this results in reduced conversion. eg. poor mixing.

Question 25

In a CSTR, the longer a molecule spends in the reactor, the ___ conversion takes place.

Answer 25

More

Question 26

In a batch/PFR reactor, the longer a molecule spends in the reactor, the ____ conversion takes place.

Answer 26

Same.

Question 27

The composition (rate) changes with ___ in a batch reactor, ____ in a PFR and ____ in a CSTR.

Answer 27

Time,
Volume (length),
Nothing.

Question 28

A CSTR operates at exit conditions so lower concentration results in ___ conversion, ___ reaction rate and ___ residence time to achieve the ___ conversion.

Answer 28

A CSTR operates at exit conditions so lower concentration results in higher conversion, lower reaction rate and longer residence time to achieve the same conversion.

Question 29

Which is more common, constant pressure or constant volume batch reactors?

Answer 29

Constant volume

Question 30

Why is space velocity used in PFRs rather than residence time?

Answer 30

Because it uses initial volumetric flow rate.

Question 31

In an ideal batch/PFR reactor, molecules have ____ residence times.

Answer 31

Equal.

Question 32

For a gas-phase reaction with constant volume, derive the residence time in a batch reactor.
n=reaction order

Answer 32

1 dt = NA0/rA V

PA = nART/V = nA0(1-xA)RT/V

rA = kPA = k nA0(1-xA)RT/V

t = 1/kRT x dxA/(1-x)

Question 33

For a gas-phase reaction, derive the residence volume occupied in a PFR.

Answer 33

1 dV = nA0/rA dxA

P = P0 (1+eAxA)
rA = kPA = k PT nA/nT 

V = nA0/k PT x [nT/nA dxA] (xA,0)

Question 34

What will happen to the mean residence time in a CSTR if teperature increases?

Answer 34

P u(t) = nRT
As T increases, u(t) increases so;
t = V/u(t)
mean residence time will decrease.

Question 35

For the following reaction A -> 2B with iner species C, when nA0=nC0=0.5, what is the nA/nT ratio?

Answer 35

nA/nT = (1-xA)/(2+xA)

Question 36

Derive the equation for the reaction rate of A in terms of residence time and concentration for a CSTR.

Answer 36

rA = (nA0-nA)/V 
rA = xA nA0/V
rA = xA CA0/(V/u)
rA = xA CA0/t
rA = (CA0 - CA)/t

Question 37

For a first order reaction, at time t, will a CSTR or PFR have a higher conversion?

Answer 37

PFR